Chapter 4: Integration

Section 4.3: Fundamental Theorem of Calculus and the Indefinite Integral

Example 4.3.4

$$ Use Maple to obtain an explicit rule for the function $G\left(x\right)\={\int }_0^x\sin \left(\sqrt{u}\right) \mathit{\ⅆ}u$; then show $G\prime \left(x\right)$ is the integrand evaluated at $x$. $$ Solution $$ •  Control-drag the equation $G\left(x\right)\=\dots$  •  Context Panel: Assign Function $G\left(x\right)\={\int }_0^x\sin \left(\sqrt{u}\right) \mathit{\ⅆ}u$$\stackrel{\text{assign as function}}{\to }$$G$ •  Write the derivative notation $G\prime \left(x\right)$ Context Panel: Evaluate and Display Inline $G\prime \left(x\right)$ = $\sin \left(\sqrt{x}\right)$$$ The explicit rule for $h\left(x\right)$ and its derivative •  Write $G\left(x\right)$ and press the Enter key. (This displays the explicit rule for $G\left(x\right)$.)   •  Context Panel: Differentiate≻With Respect To≻$x$ $G\left(x\right)$ $2\sin \left(\sqrt{x}\right)-2\sqrt{x}\cos \left(\sqrt{x}\right)$ $\stackrel{\text{differentiate w.r.t. x}}{\to }$ $\sin \left(\sqrt{x}\right)$ $$ At this point in a standard calculus course, the student would not have the seen a method for finding the explicit antiderivative $G\left(x\right)$. However, Maple can find this antiderivative, and with it, provide another illustration of the Fundamental Theorem of Calculus.$$ $$

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